--- a/src/HOL/Library/Determinants.thy Thu Mar 12 08:57:03 2009 -0700
+++ b/src/HOL/Library/Determinants.thy Thu Mar 12 09:27:23 2009 -0700
@@ -40,7 +40,7 @@
lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
- "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
+ "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
else setprod f {m..n})"
by (auto simp add: atLeastAtMostSuc_conv)
@@ -98,20 +98,20 @@
(* A few general lemmas we need below. *)
(* ------------------------------------------------------------------------- *)
-lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}"
- and i: "i \<in> {1..dimindex(UNIV::'n set)}"
+lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}"
+ and i: "i \<in> {1..dimindex(UNIV::'n set)}"
shows "Cart_nth (Cart_lambda g ::'a^'n) (p i) = g(p i)"
using permutes_in_image[OF p] i
by (simp add: Cart_lambda_beta permutes_in_image[OF p])
lemma setprod_permute:
- assumes p: "p permutes S"
+ assumes p: "p permutes S"
shows "setprod f S = setprod (f o p) S"
proof-
{assume "\<not> finite S" hence ?thesis by simp}
moreover
{assume fS: "finite S"
- then have ?thesis
+ then have ?thesis
apply (simp add: setprod_def)
apply (rule ab_semigroup_mult.fold_image_permute)
apply (auto simp add: p)
@@ -134,9 +134,9 @@
have fU: "finite ?U" by blast
{fix p assume p: "p \<in> {p. p permutes ?U}"
from p have pU: "p permutes ?U" by blast
- have sth: "sign (inv p) = sign p"
+ have sth: "sign (inv p) = sign p"
by (metis sign_inverse fU p mem_def Collect_def permutation_permutes)
- from permutes_inj[OF pU]
+ from permutes_inj[OF pU]
have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
from permutes_image[OF pU]
have "setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U = setprod (\<lambda>i. ?di (transp A) i (inv p i)) (p ` ?U)" by simp
@@ -148,7 +148,7 @@
from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
have "((\<lambda>i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)"
unfolding transp_def by (simp add: Cart_lambda_beta expand_fun_eq)}
- then show "setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
+ then show "setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
qed
finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
by simp}
@@ -156,7 +156,7 @@
apply (rule setsum_cong2) by blast
qed
-lemma det_lowerdiagonal:
+lemma det_lowerdiagonal:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i < j \<Longrightarrow> A$i$j = 0"
shows "det A = setprod (\<lambda>i. A$i$i) {1..dimindex(UNIV:: 'n set)}"
@@ -179,7 +179,7 @@
unfolding det_def by (simp add: sign_id)
qed
-lemma det_upperdiagonal:
+lemma det_upperdiagonal:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i > j \<Longrightarrow> A$i$j = 0"
shows "det A = setprod (\<lambda>i. A$i$i) {1..dimindex(UNIV:: 'n set)}"
@@ -216,7 +216,7 @@
then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_lowerdiagonal
by blast
also have "\<dots> = 1" unfolding th setprod_1 ..
- finally show ?thesis .
+ finally show ?thesis .
qed
lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
@@ -235,7 +235,7 @@
then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_lowerdiagonal
by blast
also have "\<dots> = 0" unfolding th ..
- finally show ?thesis .
+ finally show ?thesis .
qed
lemma det_permute_rows:
@@ -243,7 +243,7 @@
assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}"
shows "det(\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric] del: One_nat_def)
- apply (subst sum_permutations_compose_right[OF p])
+ apply (subst sum_permutations_compose_right[OF p])
proof(rule setsum_cong2)
let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
let ?PU = "{p. p permutes ?U}"
@@ -259,14 +259,14 @@
have "?Ap$i$ (q o p) i = A $ p i $ (q o p) i " by simp}
hence "setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = setprod (\<lambda>i. A$p i$(q o p) i) ?U"
by (auto intro: setprod_cong)
- also have "\<dots> = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
+ also have "\<dots> = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
by (simp only: setprod_permute[OF ip, symmetric])
also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
by (simp only: o_def)
also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
- finally have thp: "setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
+ finally have thp: "setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
by blast
- show "of_int (sign (q o p)) * setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
+ show "of_int (sign (q o p)) * setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
qed
@@ -282,18 +282,18 @@
moreover
have "?Ap = transp (\<chi> i. transp A $ p i)"
by (simp add: transp_def Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF p])
- ultimately show ?thesis by simp
+ ultimately show ?thesis by simp
qed
lemma det_identical_rows:
fixes A :: "'a::ordered_idom^'n^'n"
- assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
+ assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
and ij: "i \<noteq> j"
and r: "row i A = row j A"
shows "det A = 0"
proof-
- have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
+ have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
by simp
have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min)
let ?p = "Fun.swap i j id"
@@ -302,12 +302,12 @@
hence "det A = det ?A" by simp
moreover have "det A = - det ?A"
by (simp add: det_permute_rows[OF permutes_swap_id[OF i j]] sign_swap_id ij th1)
- ultimately show "det A = 0" by (metis tha)
+ ultimately show "det A = 0" by (metis tha)
qed
lemma det_identical_columns:
fixes A :: "'a::ordered_idom^'n^'n"
- assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
+ assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
and ij: "i \<noteq> j"
and r: "column i A = column j A"
@@ -316,9 +316,9 @@
apply (rule det_identical_rows[OF i j ij])
by (metis row_transp i j r)
-lemma det_zero_row:
+lemma det_zero_row:
fixes A :: "'a::{idom, ring_char_0}^'n^'n"
- assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
+ assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
and r: "row i A = 0"
shows "det A = 0"
using i r
@@ -332,16 +332,16 @@
apply (subgoal_tac "(0\<Colon>'a ^ 'n) $ a i = 0")
apply simp
apply (rule zero_index)
-apply (drule permutes_in_image[of _ _ i])
+apply (drule permutes_in_image[of _ _ i])
apply simp
-apply (drule permutes_in_image[of _ _ i])
+apply (drule permutes_in_image[of _ _ i])
apply simp
apply simp
done
lemma det_zero_column:
fixes A :: "'a::{idom,ring_char_0}^'n^'n"
- assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
+ assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
and r: "column i A = 0"
shows "det A = 0"
apply (subst det_transp[symmetric])
@@ -361,7 +361,7 @@
proof(rule setprod_cong[OF refl])
let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
fix i assume i: "i \<in> ?U"
- from Cart_lambda_beta'[OF i, of g] have
+ from Cart_lambda_beta'[OF i, of g] have
"((\<chi> i. g i) :: 'a^'n^'n) $ i = g i" .
hence "((\<chi> i. g i) :: 'a^'n^'n) $ i $ f i = g i $ f i" by simp
then
@@ -369,7 +369,7 @@
qed
lemma det_row_add:
- assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+ assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
@@ -387,7 +387,7 @@
note pin[simp] = permutes_in_image[OF pU]
have kU: "?U = insert k ?Uk" using k by blast
{fix j assume j: "j \<in> ?Uk"
- from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
+ from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
by simp_all}
then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
and th2: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?h i $ p i) ?Uk"
@@ -411,7 +411,7 @@
qed
lemma det_row_mul:
- assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+ assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
@@ -451,7 +451,7 @@
qed
lemma det_row_0:
- assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+ assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
using det_row_mul[OF k, of 0 "\<lambda>i. 1" b]
apply (simp)
@@ -483,8 +483,8 @@
let ?S = "{row j A |j. j\<in> ?U \<and> j\<noteq> i}"
let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
let ?P = "\<lambda>x. ?d (row i A + x) = det A"
- {fix k
-
+ {fix k
+
have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
then have P0: "?P 0"
apply -
@@ -499,10 +499,10 @@
apply (rule det_identical_rows[OF i j(2,3)])
using i j by (vector row_def)
have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
- then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i]
+ then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i]
by simp }
- ultimately show ?thesis
+ ultimately show ?thesis
apply -
apply (rule span_induct_alt[of ?P ?S, OF P0])
apply blast
@@ -524,7 +524,7 @@
from d obtain i where i: "i \<in> ?U" "row i A \<in> span (rows A - {row i A})"
unfolding dependent_def rows_def by blast
{fix j k assume j: "j \<in>?U" and k: "k \<in> ?U" and jk: "j \<noteq> k"
- and c: "row j A = row k A"
+ and c: "row j A = row k A"
from det_identical_rows[OF j k jk c] have ?thesis .}
moreover
{assume H: "\<And> i j. i\<in> ?U \<Longrightarrow> j \<in> ?U \<Longrightarrow> i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
@@ -537,7 +537,7 @@
from det_row_span[OF i(1) th0]
have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
unfolding right_minus vector_smult_lzero ..
- with det_row_mul[OF i(1), of "0::'a" "\<lambda>i. 1"]
+ with det_row_mul[OF i(1), of "0::'a" "\<lambda>i. 1"]
have "det A = 0" by simp}
ultimately show ?thesis by blast
qed
@@ -552,7 +552,7 @@
lemma Cart_lambda_cong: "(\<And>x. x \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
apply (rule iffD1[OF Cart_lambda_unique]) by vector
-lemma det_linear_row_setsum:
+lemma det_linear_row_setsum:
assumes fS: "finite S" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a i j else c i)::'a^'n^'n)) S"
using k
@@ -567,7 +567,7 @@
assumes fS: "finite S"
shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
proof(induct k)
- case 0
+ case 0
have th: "{f. \<forall>i. f i = i} = {id}" by (auto intro: ext)
show ?case by (auto simp add: th)
next
@@ -581,7 +581,7 @@
apply (auto intro: ext)
done
with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
- show ?case by metis
+ show ?case by metis
qed
@@ -608,9 +608,9 @@
from Suc.prems have k': "k \<le> dimindex (UNIV :: 'n set)" by arith
have thif: "\<And>a b c d. (if b \<or> a then c else d) = (if a then c else if b then c else d)" by simp
have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
- (if c then (if a then b else d) else (if a then b else e))" by simp
- have "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
- det (\<chi> i. if i = Suc k then setsum (a i) S
+ (if c then (if a then b else d) else (if a then b else e))" by simp
+ have "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
+ det (\<chi> i. if i = Suc k then setsum (a i) S
else if i \<le> k then setsum (a i) S else c i)"
unfolding le_Suc_eq thif ..
also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<le> k then setsum (a i) S
@@ -618,14 +618,14 @@
unfolding det_linear_row_setsum[OF fS Sk]
apply (subst thif2)
by (simp cong del: if_weak_cong cong add: if_cong)
- finally have tha:
- "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
+ finally have tha:
+ "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
(\<Sum>(j, f)\<in>S \<times> ?F k. det (\<chi> i. if i \<le> k then a i (f i)
else if i = Suc k then a i j
- else c i))"
+ else c i))"
unfolding Suc.hyps[OF k'] unfolding setsum_cartesian_product by blast
show ?case unfolding tha
- apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
+ apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS],
blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS], auto intro: ext)
apply (rule cong[OF refl[of det]])
@@ -637,7 +637,7 @@
shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. (\<forall>i \<in> {1 .. dimindex (UNIV :: 'n set)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1.. dimindex (UNIV :: 'n set)} \<longrightarrow> f i = i)}"
proof-
have th0: "\<And>x y. ((\<chi> i. if i <= dimindex(UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
-
+
from det_linear_rows_setsum_lemma[OF fS, of "dimindex (UNIV :: 'n set)" a, unfolded th0, OF order_refl] show ?thesis by blast
qed
@@ -674,25 +674,25 @@
have fU: "finite ?U" by simp
have fF: "finite ?F" using finite_bounded_functions[OF fU] .
{fix p assume p: "p permutes ?U"
-
+
have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
using p[unfolded permutes_def] by simp}
- then have PUF: "?PU \<subseteq> ?F" by blast
+ then have PUF: "?PU \<subseteq> ?F" by blast
{fix f assume fPU: "f \<in> ?F - ?PU"
have fUU: "f ` ?U \<subseteq> ?U" using fPU by auto
from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U"
- "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
+ "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
by auto
-
+
let ?A = "(\<chi> i. A$i$f i *s B$f i) :: 'a^'n^'n"
let ?B = "(\<chi> i. B$f i) :: 'a^'n^'n"
{assume fni: "\<not> inj_on f ?U"
then obtain i j where ij: "i \<in> ?U" "j \<in> ?U" "f i = f j" "i \<noteq> j"
unfolding inj_on_def by blast
- from ij
+ from ij
have rth: "row i ?B = row j ?B" by (vector row_def)
- from det_identical_rows[OF ij(1,2,4) rth]
- have "det (\<chi> i. A$i$f i *s B$f i) = 0"
+ from det_identical_rows[OF ij(1,2,4) rth]
+ have "det (\<chi> i. A$i$f i *s B$f i) = 0"
unfolding det_rows_mul by simp}
moreover
{assume fi: "inj_on f ?U"
@@ -701,7 +701,7 @@
apply (case_tac "i \<in> ?U")
apply (case_tac "j \<in> ?U") by metis+
note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
-
+
{fix y
from fs f have "\<exists>x. f x = y" by (cases "y \<in> ?U") blast+
then obtain x where x: "f x = y" by blast
@@ -724,17 +724,17 @@
fix q assume qU: "q \<in> ?PU"
hence q: "q permutes ?U" by blast
from p q have pp: "permutation p" and pq: "permutation q"
- unfolding permutation_permutes by auto
- have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
- "\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
- unfolding mult_assoc[symmetric] unfolding of_int_mult[symmetric]
+ unfolding permutation_permutes by auto
+ have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
+ "\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
+ unfolding mult_assoc[symmetric] unfolding of_int_mult[symmetric]
by (simp_all add: sign_idempotent)
have ths: "?s q = ?s p * ?s (q o inv p)"
using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: th00 mult_ac sign_idempotent sign_compose)
have th001: "setprod (\<lambda>i. B$i$ q (inv p i)) ?U = setprod ((\<lambda>i. B$i$ q (inv p i)) o p) ?U"
by (rule setprod_permute[OF p])
- have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
+ have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
apply (rule setprod_cong[OF refl])
using permutes_in_image[OF q] by vector
@@ -743,16 +743,16 @@
by (simp add: sign_nz th00 ring_simps sign_idempotent sign_compose)
qed
}
- then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
+ then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
unfolding det_def setsum_product
- by (rule setsum_cong2)
+ by (rule setsum_cong2)
have "det (A**B) = setsum (\<lambda>f. det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
- unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] ..
+ unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] ..
also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
- using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric]
+ using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric]
unfolding det_rows_mul by auto
finally show ?thesis unfolding th2 .
-qed
+qed
(* ------------------------------------------------------------------------- *)
(* Relation to invertibility. *)
@@ -768,7 +768,7 @@
shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). A** B = mat 1)"
by (metis invertible_def matrix_left_right_inverse)
-lemma invertible_det_nz:
+lemma invertible_det_nz:
fixes A::"real ^'n^'n"
shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
proof-
@@ -782,7 +782,7 @@
{assume H: "\<not> invertible A"
let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
have fU: "finite ?U" by simp
- from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
+ from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
unfolding invertible_righ_inverse
unfolding matrix_right_invertible_independent_rows by blast
@@ -791,14 +791,14 @@
apply (simp only: ab_left_minus add_assoc[symmetric])
apply simp
done
- from c ci
+ from c ci
have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s c j *s row j A) (?U - {i})"
- unfolding setsum_diff1'[OF fU iU] setsum_cmul
+ unfolding setsum_diff1'[OF fU iU] setsum_cmul
apply (simp add: field_simps)
apply (rule vector_mul_lcancel_imp[OF ci])
apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps)
unfolding stupid ..
- have thr: "- row i A \<in> span {row j A| j. j\<in> ?U \<and> j \<noteq> i}"
+ have thr: "- row i A \<in> span {row j A| j. j\<in> ?U \<and> j \<noteq> i}"
unfolding thr0
apply (rule span_setsum)
apply simp
@@ -808,8 +808,8 @@
apply auto
done
let ?B = "(\<chi> k. if k = i then 0 else row k A) :: real ^'n^'n"
- have thrb: "row i ?B = 0" using iU by (vector row_def)
- have "det A = 0"
+ have thrb: "row i ?B = 0" using iU by (vector row_def)
+ have "det A = 0"
unfolding det_row_span[OF iU thr, symmetric] right_minus
unfolding det_zero_row[OF iU thrb] ..}
ultimately show ?thesis by blast
@@ -823,8 +823,8 @@
fixes A:: "'a::ordered_idom^'n^'n" and x :: "'a ^'n"
assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) {1 .. dimindex(UNIV::'n set)}
- else row i A)::'a^'n^'n) = x$k * det A"
- (is "?lhs = ?rhs")
+ else row i A)::'a^'n^'n) = x$k * det A"
+ (is "?lhs = ?rhs")
proof-
let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
let ?Uk = "?U - {k}"
@@ -835,7 +835,7 @@
by (vector ring_simps)
have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by (auto intro: ext)
have "(\<chi> i. row i A) = A" by (vector row_def)
- then have thd1: "det (\<chi> i. row i A) = det A" by simp
+ then have thd1: "det (\<chi> i. row i A) = det A" by simp
have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
apply (rule det_row_span[OF k])
apply (rule span_setsum[OF fUk])
@@ -846,7 +846,7 @@
done
show "?lhs = x$k * det A"
apply (subst U)
- unfolding setsum_insert[OF fUk kUk]
+ unfolding setsum_insert[OF fUk kUk]
apply (subst th00)
unfolding add_assoc
apply (subst det_row_add[OF k])
@@ -863,8 +863,8 @@
proof-
have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transp A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
by (auto simp add: row_transp intro: setsum_cong2)
- show ?thesis
- unfolding matrix_mult_vsum
+ show ?thesis
+ unfolding matrix_mult_vsum
unfolding cramer_lemma_transp[OF k, of x "transp A", unfolded det_transp, symmetric]
unfolding stupid[of "\<lambda>i. x$i"]
apply (subst det_transp[symmetric])
@@ -873,10 +873,10 @@
lemma cramer:
fixes A ::"real^'n^'n"
- assumes d0: "det A \<noteq> 0"
+ assumes d0: "det A \<noteq> 0"
shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
proof-
- from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
+ from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def by blast
have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
@@ -896,10 +896,10 @@
lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^'n). norm (f v) = norm v)"
unfolding orthogonal_transformation_def
- apply auto
+ apply auto
apply (erule_tac x=v in allE)+
apply (simp add: real_vector_norm_def)
- by (simp add: dot_norm linear_add[symmetric])
+ by (simp add: dot_norm linear_add[symmetric])
definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transp Q ** Q = mat 1 \<and> Q ** transp Q = mat 1"
@@ -909,12 +909,12 @@
lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1)"
by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid)
-lemma orthogonal_matrix_mul:
+lemma orthogonal_matrix_mul:
fixes A :: "real ^'n^'n"
assumes oA : "orthogonal_matrix A"
- and oB: "orthogonal_matrix B"
+ and oB: "orthogonal_matrix B"
shows "orthogonal_matrix(A ** B)"
- using oA oB
+ using oA oB
unfolding orthogonal_matrix matrix_transp_mul
apply (subst matrix_mul_assoc)
apply (subst matrix_mul_assoc[symmetric])
@@ -939,7 +939,7 @@
"\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
by simp_all
from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul] i j
- have "?A$i$j = ?m1 $ i $ j"
+ have "?A$i$j = ?m1 $ i $ j"
by (simp add: Cart_lambda_beta' dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def del: One_nat_def)}
hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
with lf have ?rhs by blast}
@@ -953,17 +953,17 @@
ultimately show ?thesis by blast
qed
-lemma det_orthogonal_matrix:
+lemma det_orthogonal_matrix:
fixes Q:: "'a::ordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
proof-
-
- have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
- proof-
+
+ have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
+ proof-
fix x:: 'a
have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: ring_simps)
- have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
+ have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
apply (subst eq_iff_diff_eq_0) by simp
have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
@@ -972,27 +972,27 @@
from oQ have "Q ** transp Q = mat 1" by (metis orthogonal_matrix_def)
hence "det (Q ** transp Q) = det (mat 1:: 'a^'n^'n)" by simp
hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transp)
- then show ?thesis unfolding th .
+ then show ?thesis unfolding th .
qed
(* ------------------------------------------------------------------------- *)
(* Linearity of scaling, and hence isometry, that preserves origin. *)
(* ------------------------------------------------------------------------- *)
-lemma scaling_linear:
+lemma scaling_linear:
fixes f :: "real ^'n \<Rightarrow> real ^'n"
assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
shows "linear f"
proof-
- {fix v w
+ {fix v w
{fix x note fd[rule_format, of x 0, unfolded dist_def f0 diff_0_right] }
note th0 = this
- have "f v \<bullet> f w = c^2 * (v \<bullet> w)"
+ have "f v \<bullet> f w = c^2 * (v \<bullet> w)"
unfolding dot_norm_neg dist_def[symmetric]
unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
note fc = this
show ?thesis unfolding linear_def vector_eq
by (simp add: dot_lmult dot_ladd dot_rmult dot_radd fc ring_simps)
-qed
+qed
lemma isometry_linear:
"f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
@@ -1005,7 +1005,7 @@
lemma orthogonal_transformation_isometry:
"orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
- unfolding orthogonal_transformation
+ unfolding orthogonal_transformation
apply (rule iffI)
apply clarify
apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_def)
@@ -1028,12 +1028,12 @@
and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
proof-
- {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
+ {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
assume H: "x = norm x *s x0" "y = norm y *s y0"
- "x' = norm x *s x0'" "y' = norm y *s y0'"
+ "x' = norm x *s x0'" "y' = norm y *s y0'"
"norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
"norm(x0' - y0') = norm(x0 - y0)"
-
+
have "norm(x' - y') = norm(x - y)"
apply (subst H(1))
apply (subst H(2))
@@ -1055,13 +1055,13 @@
then have "dist (?g x) (?g y) = dist x y" by simp }
moreover
{assume "x = 0" "y \<noteq> 0"
- then have "dist (?g x) (?g y) = dist x y"
+ then have "dist (?g x) (?g y) = dist x y"
apply (simp add: dist_def norm_mul)
apply (rule f1[rule_format])
by(simp add: norm_mul field_simps)}
moreover
{assume "x \<noteq> 0" "y = 0"
- then have "dist (?g x) (?g y) = dist x y"
+ then have "dist (?g x) (?g y) = dist x y"
apply (simp add: dist_def norm_mul)
apply (rule f1[rule_format])
by(simp add: norm_mul field_simps)}
@@ -1077,14 +1077,14 @@
norm (inverse (norm x) *s x - inverse (norm y) *s y)"
using z
by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
- from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
+ from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
by (simp add: dist_def)}
ultimately have "dist (?g x) (?g y) = dist x y" by blast}
note thd = this
- show ?thesis
+ show ?thesis
apply (rule exI[where x= ?g])
unfolding orthogonal_transformation_isometry
- using g0 thfg thd by metis
+ using g0 thfg thd by metis
qed
(* ------------------------------------------------------------------------- *)
@@ -1094,7 +1094,7 @@
definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
-lemma orthogonal_rotation_or_rotoinversion:
+lemma orthogonal_rotation_or_rotoinversion:
fixes Q :: "'a::ordered_idom^'n^'n"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
@@ -1104,9 +1104,9 @@
lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
-lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
+lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
by (simp add: nat_number setprod_numseg mult_commute)
-lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
+lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
by (simp add: nat_number setprod_numseg mult_commute)
lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
@@ -1116,7 +1116,7 @@
proof-
have f12: "finite {2::nat}" "1 \<notin> {2::nat}" by auto
have th12: "{1 .. 2} = insert (1::nat) {2}" by auto
- show ?thesis
+ show ?thesis
apply (simp add: det_def dimindex_def th12 del: One_nat_def)
unfolding setsum_over_permutations_insert[OF f12]
unfolding permutes_sing
@@ -1124,7 +1124,7 @@
by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
qed
-lemma det_3: "det (A::'a::comm_ring_1^3^3) =
+lemma det_3: "det (A::'a::comm_ring_1^3^3) =
A$1$1 * A$2$2 * A$3$3 +
A$1$2 * A$2$3 * A$3$1 +
A$1$3 * A$2$1 * A$3$2 -
@@ -1136,7 +1136,7 @@
have f23: "finite {(3::nat)}" "2 \<notin> {(3::nat)}" by auto
have th12: "{1 .. 3} = insert (1::nat) (insert 2 {3})" by auto
- show ?thesis
+ show ?thesis
apply (simp add: det_def dimindex_def th12 del: One_nat_def)
unfolding setsum_over_permutations_insert[OF f123]
unfolding setsum_over_permutations_insert[OF f23]
--- a/src/HOL/Library/Euclidean_Space.thy Thu Mar 12 08:57:03 2009 -0700
+++ b/src/HOL/Library/Euclidean_Space.thy Thu Mar 12 09:27:23 2009 -0700
@@ -5,7 +5,7 @@
header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
theory Euclidean_Space
- imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main
+ imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main
Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
Inner_Product
uses ("normarith.ML")
@@ -31,26 +31,26 @@
qed
lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
-lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1"
+lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1"
by (simp add: atLeastAtMost_singleton)
-lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2"
+lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2"
by (simp add: nat_number atLeastAtMostSuc_conv add_commute)
-lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3"
+lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3"
by (simp add: nat_number atLeastAtMostSuc_conv add_commute)
subsection{* Basic componentwise operations on vectors. *}
instantiation "^" :: (plus,type) plus
begin
-definition vector_add_def : "op + \<equiv> (\<lambda> x y. (\<chi> i. (x$i) + (y$i)))"
+definition vector_add_def : "op + \<equiv> (\<lambda> x y. (\<chi> i. (x$i) + (y$i)))"
instance ..
end
instantiation "^" :: (times,type) times
begin
- definition vector_mult_def : "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
+ definition vector_mult_def : "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
instance ..
end
@@ -64,12 +64,12 @@
instance ..
end
instantiation "^" :: (zero,type) zero begin
- definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
+ definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
instance ..
end
instantiation "^" :: (one,type) one begin
- definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
+ definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
instance ..
end
@@ -80,13 +80,13 @@
x$i <= y$i)"
definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
dimindex (UNIV :: 'b set)}. x$i < y$i)"
-
+
instance by (intro_classes)
end
instantiation "^" :: (scaleR, type) scaleR
begin
-definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
+definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
instance ..
end
@@ -117,19 +117,19 @@
lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
method_setup vector = {*
let
- val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
- @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
+ val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
+ @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
@{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
- val ss2 = @{simpset} addsimps
- [@{thm vector_add_def}, @{thm vector_mult_def},
- @{thm vector_minus_def}, @{thm vector_uminus_def},
- @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
+ val ss2 = @{simpset} addsimps
+ [@{thm vector_add_def}, @{thm vector_mult_def},
+ @{thm vector_minus_def}, @{thm vector_uminus_def},
+ @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
@{thm vector_scaleR_def},
@{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}]
- fun vector_arith_tac ths =
+ fun vector_arith_tac ths =
simp_tac ss1
THEN' (fn i => rtac @{thm setsum_cong2} i
- ORELSE rtac @{thm setsum_0'} i
+ ORELSE rtac @{thm setsum_0'} i
ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
(* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
THEN' asm_full_simp_tac (ss2 addsimps ths)
@@ -145,30 +145,30 @@
text{* Obvious "component-pushing". *}
-lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x"
- by (vector vec_def)
-
-lemma vector_add_component:
+lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x"
+ by (vector vec_def)
+
+lemma vector_add_component:
fixes x y :: "'a::{plus} ^ 'n" assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(x + y)$i = x$i + y$i"
using i by vector
-lemma vector_minus_component:
+lemma vector_minus_component:
fixes x y :: "'a::{minus} ^ 'n" assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(x - y)$i = x$i - y$i"
using i by vector
-lemma vector_mult_component:
+lemma vector_mult_component:
fixes x y :: "'a::{times} ^ 'n" assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(x * y)$i = x$i * y$i"
using i by vector
-lemma vector_smult_component:
+lemma vector_smult_component:
fixes y :: "'a::{times} ^ 'n" assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(c *s y)$i = c * (y$i)"
using i by vector
-lemma vector_uminus_component:
+lemma vector_uminus_component:
fixes x :: "'a::{uminus} ^ 'n" assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(- x)$i = - (x$i)"
using i by vector
@@ -188,26 +188,26 @@
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
-instance "^" :: (semigroup_add,type) semigroup_add
+instance "^" :: (semigroup_add,type) semigroup_add
apply (intro_classes) by (vector add_assoc)
-instance "^" :: (monoid_add,type) monoid_add
- apply (intro_classes) by vector+
-
-instance "^" :: (group_add,type) group_add
- apply (intro_classes) by (vector algebra_simps)+
-
-instance "^" :: (ab_semigroup_add,type) ab_semigroup_add
+instance "^" :: (monoid_add,type) monoid_add
+ apply (intro_classes) by vector+
+
+instance "^" :: (group_add,type) group_add
+ apply (intro_classes) by (vector algebra_simps)+
+
+instance "^" :: (ab_semigroup_add,type) ab_semigroup_add
apply (intro_classes) by (vector add_commute)
instance "^" :: (comm_monoid_add,type) comm_monoid_add
apply (intro_classes) by vector
-instance "^" :: (ab_group_add,type) ab_group_add
+instance "^" :: (ab_group_add,type) ab_group_add
apply (intro_classes) by vector+
-instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add
+instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add
apply (intro_classes)
by (vector Cart_eq)+
@@ -218,30 +218,30 @@
instance "^" :: (real_vector, type) real_vector
by default (vector scaleR_left_distrib scaleR_right_distrib)+
-instance "^" :: (semigroup_mult,type) semigroup_mult
+instance "^" :: (semigroup_mult,type) semigroup_mult
apply (intro_classes) by (vector mult_assoc)
-instance "^" :: (monoid_mult,type) monoid_mult
+instance "^" :: (monoid_mult,type) monoid_mult
apply (intro_classes) by vector+
-instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
+instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
apply (intro_classes) by (vector mult_commute)
-instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
+instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
apply (intro_classes) by (vector mult_idem)
-instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
+instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
apply (intro_classes) by vector
fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
"vector_power x 0 = 1"
| "vector_power x (Suc n) = x * vector_power x n"
-instantiation "^" :: (recpower,type) recpower
+instantiation "^" :: (recpower,type) recpower
begin
definition vec_power_def: "op ^ \<equiv> vector_power"
- instance
- apply (intro_classes) by (simp_all add: vec_power_def)
+ instance
+ apply (intro_classes) by (simp_all add: vec_power_def)
end
instance "^" :: (semiring,type) semiring
@@ -250,16 +250,16 @@
instance "^" :: (semiring_0,type) semiring_0
apply (intro_classes) by (vector ring_simps)+
instance "^" :: (semiring_1,type) semiring_1
- apply (intro_classes) apply vector using dimindex_ge_1 by auto
+ apply (intro_classes) apply vector using dimindex_ge_1 by auto
instance "^" :: (comm_semiring,type) comm_semiring
apply (intro_classes) by (vector ring_simps)+
-instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
+instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
-instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
-instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
-instance "^" :: (ring,type) ring by (intro_classes)
-instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
+instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
+instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
+instance "^" :: (ring,type) ring by (intro_classes)
+instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
instance "^" :: (ring_1,type) ring_1 ..
@@ -273,31 +273,31 @@
instance "^" :: (real_algebra_1,type) real_algebra_1 ..
-lemma of_nat_index:
+lemma of_nat_index:
"i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
apply (induct n)
apply vector
apply vector
done
-lemma zero_index[simp]:
+lemma zero_index[simp]:
"i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
-lemma one_index[simp]:
+lemma one_index[simp]:
"i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
proof-
have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
- also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
- finally show ?thesis by simp
+ also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
+ finally show ?thesis by simp
qed
-instance "^" :: (semiring_char_0,type) semiring_char_0
-proof (intro_classes)
+instance "^" :: (semiring_char_0,type) semiring_char_0
+proof (intro_classes)
fix m n ::nat
show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
proof(induct m arbitrary: n)
- case 0 thus ?case apply vector
+ case 0 thus ?case apply vector
apply (induct n,auto simp add: ring_simps)
using dimindex_ge_1 apply auto
apply vector
@@ -323,24 +323,24 @@
instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
-lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
+lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
by (vector mult_assoc)
-lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
+lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
by (vector ring_simps)
-lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
+lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
by (vector ring_simps)
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
-lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
+lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
by (vector ring_simps)
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
-lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
+lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
by (vector ring_simps)
-lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
+lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
using dimindex_ge_1 apply auto done
@@ -581,15 +581,15 @@
subsection{* Properties of the dot product. *}
-lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
+lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
by (vector mult_commute)
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
by (vector ring_simps)
-lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
+lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
by (vector ring_simps)
-lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
+lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
by (vector ring_simps)
-lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
+lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
by (vector ring_simps)
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
@@ -625,8 +625,8 @@
ultimately show ?thesis by metis
qed
-lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
- by (auto simp add: le_less)
+lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
+ by (auto simp add: le_less)
subsection{* The collapse of the general concepts to dimension one. *}
@@ -642,13 +642,13 @@
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
by (simp add: vector_norm_def dimindex_def)
-lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
+lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
by (simp add: norm_vector_1)
text{* Metric *}
text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
-definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where
+definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where
"dist x y = norm (x - y)"
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
@@ -667,14 +667,14 @@
shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
proof-
let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
- have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
- have Sub: "\<exists>y. isUb UNIV ?S y"
+ have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
+ have Sub: "\<exists>y. isUb UNIV ?S y"
apply (rule exI[where x= b])
- using ab fb e12 by (auto simp add: isUb_def setle_def)
- from reals_complete[OF Se Sub] obtain l where
+ using ab fb e12 by (auto simp add: isUb_def setle_def)
+ from reals_complete[OF Se Sub] obtain l where
l: "isLub UNIV ?S l"by blast
have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
- apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
+ apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
by (metis linorder_linear)
have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
@@ -685,11 +685,11 @@
{assume le2: "f l \<in> e2"
from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
hence lap: "l - a > 0" using alb by arith
- from e2[rule_format, OF le2] obtain e where
+ from e2[rule_format, OF le2] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
- from dst[OF alb e(1)] obtain d where
+ from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
- have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
+ have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
apply ferrack by arith
then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
@@ -701,16 +701,16 @@
{assume le1: "f l \<in> e1"
from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
hence blp: "b - l > 0" using alb by arith
- from e1[rule_format, OF le1] obtain e where
+ from e1[rule_format, OF le1] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
- from dst[OF alb e(1)] obtain d where
+ from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
- have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
+ have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
then obtain d' where d': "d' > 0" "d' < d" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
- with l d' have False
+ with l d' have False
by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
ultimately show ?thesis using alb by metis
qed
@@ -719,7 +719,7 @@
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
proof-
- have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
+ have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
thus ?thesis by (simp add: ring_simps power2_eq_square)
qed
@@ -740,14 +740,14 @@
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
using real_sqrt_less_mono[of "x^2" y] by simp
-lemma sqrt_even_pow2: assumes n: "even n"
+lemma sqrt_even_pow2: assumes n: "even n"
shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
proof-
- from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
- by (auto simp add: nat_number)
+ from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
+ by (auto simp add: nat_number)
from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
by (simp only: power_mult[symmetric] mult_commute)
- then show ?thesis using m by simp
+ then show ?thesis using m by simp
qed
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
@@ -786,7 +786,7 @@
{assume "norm x = 0"
hence ?thesis by (simp add: dot_lzero dot_rzero)}
moreover
- {assume "norm y = 0"
+ {assume "norm y = 0"
hence ?thesis by (simp add: dot_lzero dot_rzero)}
moreover
{assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
@@ -829,7 +829,7 @@
lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
by (simp add: vector_norm_def setL2_le_setsum)
-lemma real_abs_norm[simp]: "\<bar> norm x\<bar> = norm (x :: real ^'n)"
+lemma real_abs_norm[simp]: "\<bar> norm x\<bar> = norm (x :: real ^'n)"
by (rule abs_norm_cancel)
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
by (rule norm_triangle_ineq3)
@@ -863,7 +863,7 @@
apply arith
done
-lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
+lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
@@ -891,7 +891,7 @@
next
assume ?rhs
then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
- hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
+ hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
by (simp add: dot_rsub dot_lsub dot_sym)
then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
then show "x = y" by (simp add: dot_eq_0)
@@ -919,13 +919,13 @@
lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
-lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
-lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
+lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
+lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
"c *s x + (d *s x + z) == (c + d) *s x + z"
"(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
-lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
+lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
"(c *s x + z) + d *s y == c *s x + (z + d *s y)"
"c *s x + (d *s y + z) == c *s x + (d *s y + z)"
"(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
@@ -941,7 +941,7 @@
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
-lemma norm_pths:
+lemma norm_pths:
"(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
"x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
using norm_ge_zero[of "x - y"] by auto
@@ -967,26 +967,26 @@
lemma dist_eq_0[simp]: "dist x y = 0 \<longleftrightarrow> x = y" by norm
-lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm
-lemma dist_nz: "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm
-
-lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm
-
-lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm
-
-lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm
-
-lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm
+lemma dist_pos_lt: "x \<noteq> y ==> 0 < dist x y" by norm
+lemma dist_nz: "x \<noteq> y \<longleftrightarrow> 0 < dist x y" by norm
+
+lemma dist_triangle_le: "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e" by norm
+
+lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm
+
+lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm
+
+lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm
lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'"
- by norm
-
-lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
- unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul ..
-
-lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm
-
-lemma dist_le_0[simp]: "dist x y <= 0 \<longleftrightarrow> x = y" by norm
+ by norm
+
+lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
+ unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul ..
+
+lemma dist_triangle_add_half: " dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm
+
+lemma dist_le_0[simp]: "dist x y <= 0 \<longleftrightarrow> x = y" by norm
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
apply vector
@@ -996,24 +996,24 @@
apply (auto simp add: vector_component zero_index)
done
-lemma setsum_clauses:
+lemma setsum_clauses:
shows "setsum f {} = 0"
and "finite S \<Longrightarrow> setsum f (insert x S) =
(if x \<in> S then setsum f S else f x + setsum f S)"
by (auto simp add: insert_absorb)
-lemma setsum_cmul:
+lemma setsum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
-lemma setsum_component:
+lemma setsum_component:
fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
using i by (simp add: setsum_eq Cart_lambda_beta)
-lemma setsum_norm:
+lemma setsum_norm:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
@@ -1027,7 +1027,7 @@
finally show ?case using "2.hyps" by simp
qed
-lemma real_setsum_norm:
+lemma real_setsum_norm:
fixes f :: "'a \<Rightarrow> real ^'n"
assumes fS: "finite S"
shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
@@ -1041,25 +1041,25 @@
finally show ?case using "2.hyps" by simp
qed
-lemma setsum_norm_le:
+lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
proof-
- from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
+ from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
by - (rule setsum_mono, simp)
then show ?thesis using setsum_norm[OF fS, of f] fg
by arith
qed
-lemma real_setsum_norm_le:
+lemma real_setsum_norm_le:
fixes f :: "'a \<Rightarrow> real ^ 'n"
assumes fS: "finite S"
and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
proof-
- from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
+ from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
by - (rule setsum_mono, simp)
then show ?thesis using real_setsum_norm[OF fS, of f] fg
by arith
@@ -1089,9 +1089,9 @@
case 1 then show ?case by (simp add: vector_smult_lzero)
next
case (2 x F)
- from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
+ from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
by simp
- also have "\<dots> = f x *s v + setsum f F *s v"
+ also have "\<dots> = f x *s v + setsum f F *s v"
by (simp add: vector_sadd_rdistrib)
also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
finally show ?case .
@@ -1105,20 +1105,20 @@
proof-
let ?A = "{m .. n}"
let ?B = "{n + 1 .. n + p}"
- have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
+ have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
have d: "?A \<inter> ?B = {}" by auto
from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
qed
lemma setsum_natinterval_left:
- assumes mn: "(m::nat) <= n"
+ assumes mn: "(m::nat) <= n"
shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
proof-
from mn have "{m .. n} = insert m {m+1 .. n}" by auto
then show ?thesis by auto
qed
-lemma setsum_natinterval_difff:
+lemma setsum_natinterval_difff:
fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
shows "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
(if m <= n then f m - f(n + 1) else 0)"
@@ -1136,8 +1136,8 @@
proof-
{fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
note th0 = this
- have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
- apply (rule setsum_cong2)
+ have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
+ apply (rule setsum_cong2)
by (simp add: th0)
also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
apply (rule setsum_setsum_restrict[OF fS])
@@ -1149,14 +1149,14 @@
lemma setsum_group:
assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
-
+
apply (subst setsum_image_gen[OF fS, of g f])
apply (rule setsum_mono_zero_right[OF fT fST])
by (auto intro: setsum_0')
lemma vsum_norm_allsubsets_bound:
fixes f:: "'a \<Rightarrow> real ^'n"
- assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
+ assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) * e"
proof-
let ?d = "real (dimindex (UNIV ::'n set))"
@@ -1183,9 +1183,9 @@
have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"] fPs[OF PnP]
by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
- have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
+ have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
apply (subst thp)
- apply (rule setsum_Un_zero)
+ apply (rule setsum_Un_zero)
using fP thp0 by auto
also have "\<dots> \<le> 2*e" using Pne Ppe by arith
finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
@@ -1204,13 +1204,13 @@
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
-lemma delta_mult_idempotent:
+lemma delta_mult_idempotent:
"(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
lemma norm_basis:
assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
shows "norm (basis k :: real ^'n) = 1"
- using k
+ using k
apply (simp add: basis_def real_vector_norm_def dot_def)
apply (vector delta_mult_idempotent)
using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
@@ -1228,7 +1228,7 @@
apply (rule exI[where x="c *s basis 1"])
by (simp only: norm_mul norm_basis_1)
-lemma vector_choose_dist: assumes e: "0 <= e"
+lemma vector_choose_dist: assumes e: "0 <= e"
shows "\<exists>(y::real^'n). dist x y = e"
proof-
from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
@@ -1250,7 +1250,7 @@
"setsum (\<lambda>i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
-lemma basis_expansion_unique:
+lemma basis_expansion_unique:
"setsum (\<lambda>i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)"
by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
@@ -1266,7 +1266,7 @@
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
by (auto simp add: Cart_eq basis_component zero_index)
-lemma basis_nonzero:
+lemma basis_nonzero:
assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
using k by (simp add: basis_eq_0)
@@ -1294,15 +1294,15 @@
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
lemma orthogonal_basis:
- assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
+ assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
using i
by (auto simp add: orthogonal_def dot_def basis_def Cart_lambda_beta cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
lemma orthogonal_basis_basis:
- assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
- and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}"
- shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j"
+ assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
+ and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}"
+ shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j"
unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
(* FIXME : Maybe some of these require less than comm_ring, but not all*)
@@ -1443,14 +1443,14 @@
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
unfolding vector_sneg_minus1
- using linear_cmul[of f] by auto
-
-lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
+ using linear_cmul[of f] by auto
+
+lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
by (simp add: diff_def linear_add linear_neg)
-lemma linear_setsum:
+lemma linear_setsum:
fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum g S) = setsum (f o g) S"
@@ -1470,7 +1470,7 @@
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
- linear_cmul[OF lf] by simp
+ linear_cmul[OF lf] by simp
lemma linear_injective_0:
assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
@@ -1478,7 +1478,7 @@
proof-
have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
- also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
+ also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
by (simp add: linear_sub[OF lf])
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
finally show ?thesis .
@@ -1518,7 +1518,7 @@
assumes lf: "linear f"
shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
proof-
- from linear_bounded[OF lf] obtain B where
+ from linear_bounded[OF lf] obtain B where
B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
@@ -1562,15 +1562,15 @@
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
using add_imp_eq[of x y 0] by auto
-
-lemma bilinear_lzero:
+
+lemma bilinear_lzero:
fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
- using bilinear_ladd[OF bh, of 0 0 x]
+ using bilinear_ladd[OF bh, of 0 0 x]
by (simp add: eq_add_iff ring_simps)
-lemma bilinear_rzero:
+lemma bilinear_rzero:
fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
- using bilinear_radd[OF bh, of x 0 0 ]
+ using bilinear_radd[OF bh, of x 0 0 ]
by (simp add: eq_add_iff ring_simps)
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
@@ -1583,7 +1583,7 @@
fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
-proof-
+proof-
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
apply (rule linear_setsum[unfolded o_def])
using bh fS by (auto simp add: bilinear_def)
@@ -1598,7 +1598,7 @@
fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof-
+proof-
let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
@@ -1626,7 +1626,7 @@
assumes bh: "bilinear h"
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof-
- from bilinear_bounded[OF bh] obtain B where
+ from bilinear_bounded[OF bh] obtain B where
B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
@@ -1634,11 +1634,11 @@
{fix x::"real ^'m" and y :: "real ^'n"
from KB Kp
have "B * norm x * norm y \<le> ?K * norm x * norm y"
- apply -
+ apply -
apply (rule mult_right_mono, rule mult_right_mono)
by (auto simp add: norm_ge_zero)
then have "norm (h x y) \<le> ?K * norm x * norm y"
- using B[rule_format, of x y] by simp}
+ using B[rule_format, of x y] by simp}
with Kp show ?thesis by blast
qed
@@ -1663,14 +1663,14 @@
have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
by (simp only: basis_expansion)
also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
- unfolding linear_setsum[OF lf fN]
+ unfolding linear_setsum[OF lf fN]
by (simp add: linear_cmul[OF lf])
finally have "f x \<bullet> y = x \<bullet> ?w"
apply (simp only: )
apply (simp add: dot_def setsum_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)
done}
}
- then show ?thesis unfolding adjoint_def
+ then show ?thesis unfolding adjoint_def
some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
by metis
@@ -1715,27 +1715,27 @@
consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
-defs (overloaded)
+defs (overloaded)
matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
-abbreviation
+abbreviation
matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" (infixl "**" 70)
where "m ** m' == m\<star> m'"
-defs (overloaded)
+defs (overloaded)
matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
-abbreviation
+abbreviation
matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" (infixl "*v" 70)
- where
+ where
"m *v v == m \<star> v"
-defs (overloaded)
+defs (overloaded)
vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
-abbreviation
+abbreviation
vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " (infixl "v*" 70)
- where
+ where
"v v* m == v \<star> m"
definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
@@ -1749,11 +1749,11 @@
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
-lemma setsum_delta':
- assumes fS: "finite S" shows
- "setsum (\<lambda>k. if a = k then b k else 0) S =
+lemma setsum_delta':
+ assumes fS: "finite S" shows
+ "setsum (\<lambda>k. if a = k then b k else 0) S =
(if a\<in> S then b a else 0)"
- using setsum_delta[OF fS, of a b, symmetric]
+ using setsum_delta[OF fS, of a b, symmetric]
by (auto intro: setsum_cong)
lemma matrix_mul_lid: "mat 1 ** A = A"
@@ -1781,7 +1781,7 @@
lemma matrix_vector_mul_lid: "mat 1 *v x = x"
apply (vector matrix_vector_mult_def mat_def)
- by (simp add: cond_value_iff cond_application_beta
+ by (simp add: cond_value_iff cond_application_beta
setsum_delta' cong del: if_weak_cong)
lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
@@ -1796,7 +1796,7 @@
apply (erule_tac x="i" in ballE)
by (auto simp add: basis_def cond_value_iff cond_application_beta Cart_lambda_beta setsum_delta[OF finite_atLeastAtMost] cong del: if_weak_cong)
-lemma matrix_vector_mul_component:
+lemma matrix_vector_mul_component:
assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
using k
@@ -1813,18 +1813,18 @@
lemma transp_transp: "transp(transp A) = A"
by (vector transp_def)
-lemma row_transp:
+lemma row_transp:
fixes A:: "'a::semiring_1^'n^'m"
assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
shows "row i (transp A) = column i A"
- using i
+ using i
by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
lemma column_transp:
fixes A:: "'a::semiring_1^'n^'m"
assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
shows "column i (transp A) = row i A"
- using i
+ using i
by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
@@ -1890,8 +1890,8 @@
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
-lemma matrix_compose:
- assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g"
+lemma matrix_compose:
+ assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g"
shows "matrix (g o f) = matrix g ** matrix f"
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
@@ -1923,9 +1923,9 @@
done
-lemma real_convex_bound_lt:
+lemma real_convex_bound_lt:
assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
- and uv: "u + v = 1"
+ and uv: "u + v = 1"
shows "u * x + v * y < a"
proof-
have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
@@ -1937,7 +1937,7 @@
apply (cases "u = 0", simp_all add: uv')
apply(rule mult_strict_left_mono)
using uv' apply simp_all
-
+
apply (rule add_less_le_mono)
apply(rule mult_strict_left_mono)
apply simp_all
@@ -1947,9 +1947,9 @@
thus ?thesis unfolding th .
qed
-lemma real_convex_bound_le:
+lemma real_convex_bound_le:
assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
- and uv: "u + v = 1"
+ and uv: "u + v = 1"
shows "u * x + v * y \<le> a"
proof-
from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
@@ -1969,7 +1969,7 @@
done
-lemma triangle_lemma:
+lemma triangle_lemma:
assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
shows "x <= y + z"
proof-
@@ -1992,12 +1992,12 @@
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
{fix i assume i: "i \<in> ?S"
with f i have "P i (f i)" by metis
- then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto
+ then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto
}
hence "\<forall>i \<in> ?S. P i (?x$i)" by metis
hence ?rhs by metis }
ultimately show ?thesis by metis
-qed
+qed
(* Supremum and infimum of real sets *)
@@ -2019,7 +2019,7 @@
lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
proof-
from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
- from rsup[OF Se] Sb have "isLub UNIV S (rsup S)" by blast
+ from rsup[OF Se] Sb have "isLub UNIV S (rsup S)" by blast
then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
qed
@@ -2030,12 +2030,12 @@
let ?m = "Max S"
from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
- from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
+ from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
- moreover
+ moreover
have "rsup S \<le> ?m" using Sm lub
by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
- ultimately show ?thesis by arith
+ ultimately show ?thesis by arith
qed
lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
@@ -2065,7 +2065,7 @@
lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
shows "rsup S = b"
-using b S
+using b S
unfolding setle_def rsup_alt
apply -
apply (rule some_equality)
@@ -2104,7 +2104,7 @@
lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
proof-
have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
- show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
+ show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
by (auto simp add: setge_def setle_def)
qed
@@ -2142,7 +2142,7 @@
lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
proof-
from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
- from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)" by blast
+ from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)" by blast
then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
qed
@@ -2153,12 +2153,12 @@
let ?m = "Min S"
from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
- from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
+ from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
- moreover
+ moreover
have "rinf S \<ge> ?m" using Sm glb
by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
- ultimately show ?thesis by arith
+ ultimately show ?thesis by arith
qed
lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
@@ -2188,7 +2188,7 @@
lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
shows "rinf S = b"
-using b S
+using b S
unfolding setge_def rinf_alt
apply -
apply (rule some_equality)
@@ -2226,7 +2226,7 @@
lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"
proof-
have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
- show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
+ show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
by (auto simp add: setge_def setle_def)
qed
@@ -2248,7 +2248,7 @@
moreover
{assume H: ?lhs
- from H[rule_format, of "basis 1"]
+ from H[rule_format, of "basis 1"]
have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
{fix x :: "real ^'n"
@@ -2260,9 +2260,9 @@
let ?c = "1/ norm x"
have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
with H have "norm (f(?c*s x)) \<le> b" by blast
- hence "?c * norm (f x) \<le> b"
+ hence "?c * norm (f x) \<le> b"
by (simp add: linear_cmul[OF lf] norm_mul)
- hence "norm (f x) \<le> b * norm x"
+ hence "norm (f x) \<le> b * norm x"
using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
ultimately have "norm (f x) \<le> b * norm x" by blast}
then have ?rhs by blast}
@@ -2278,16 +2278,16 @@
{
let ?S = "{norm (f x) |x. norm x = 1}"
have Se: "?S \<noteq> {}" using norm_basis_1 by auto
- from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
+ from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
{from rsup[OF Se b, unfolded onorm_def[symmetric]]
- show "norm (f x) <= onorm f * norm x"
- apply -
+ show "norm (f x) <= onorm f * norm x"
+ apply -
apply (rule spec[where x = x])
unfolding norm_bound_generalize[OF lf, symmetric]
by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
{
- show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
+ show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
using rsup[OF Se b, unfolded onorm_def[symmetric]]
unfolding norm_bound_generalize[OF lf, symmetric]
by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
@@ -2297,7 +2297,7 @@
lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
-lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
+lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
using onorm[OF lf]
apply (auto simp add: onorm_pos_le)
@@ -2317,7 +2317,7 @@
apply (rule rsup_unique) by (simp_all add: setle_def)
qed
-lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
+lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
unfolding onorm_eq_0[OF lf, symmetric]
using onorm_pos_le[OF lf] by arith
@@ -2374,7 +2374,7 @@
definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
-definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
+definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
where "dest_vec1 x = (x$1)"
lemma vec1_component[simp]: "(vec1 x)$1 = x"
@@ -2385,7 +2385,7 @@
lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
-lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
+lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))" by (metis vec1_dest_vec1)
@@ -2446,7 +2446,7 @@
lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
by (metis vec1_dest_vec1 norm_vec1)
-lemma linear_vmul_dest_vec1:
+lemma linear_vmul_dest_vec1:
fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
unfolding dest_vec1_def
@@ -2563,10 +2563,10 @@
have th_0: "1 \<le> ?n +1" by simp
have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
by (simp add: pastecart_fst_snd)
- have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
+ have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square del: One_nat_def)
then show ?thesis
- unfolding th0
+ unfolding th0
unfolding real_vector_norm_def real_sqrt_le_iff id_def
by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
qed
@@ -2592,13 +2592,13 @@
using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"]
apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def)
by arith
- have fS: "?f ` ?S = ?M"
+ have fS: "?f ` ?S = ?M"
apply (rule set_ext)
apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith
- have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
- by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)
+ have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
+ by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)
then show ?thesis
- unfolding th0
+ unfolding th0
unfolding real_vector_norm_def real_sqrt_le_iff id_def
by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
qed
@@ -2644,14 +2644,14 @@
done
let ?r = "\<lambda>n. n - ?n"
have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith
- have rS: "?r ` ?S = ?M" apply (rule set_ext)
+ have rS: "?r ` ?S = ?M" apply (rule set_ext)
apply (simp add: thnm image_iff Bex_def) by arith
have "pastecart x1 x2 \<bullet> (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def)
also have "\<dots> = setsum ?g ?N + setsum ?g ?S"
by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def)
also have "\<dots> = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M"
unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 ..
- finally
+ finally
show ?thesis by (simp add: dot_def)
qed
@@ -2679,7 +2679,7 @@
unfolding hull_def subset_iff by auto
lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
-using hull_same[of s S] hull_in[of S s] by metis
+using hull_same[of s S] hull_in[of S s] by metis
lemma hull_hull: "S hull (S hull s) = S hull s"
@@ -2749,12 +2749,12 @@
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
proof(induct n)
case 0 thus ?case by simp
-next
+next
case (Suc n)
hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
- also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
+ also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
apply (simp add: ring_simps)
using mult_left_mono[OF p Suc.prems] by simp
finally show ?case by (simp add: real_of_nat_Suc ring_simps)
@@ -2763,13 +2763,13 @@
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
proof-
from x have x0: "x - 1 > 0" by arith
- from real_arch[OF x0, rule_format, of y]
+ from real_arch[OF x0, rule_format, of y]
obtain n::nat where n:"y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 \<ge> 0" by arith
- from real_pow_lbound[OF x00, of n] n
+ from real_pow_lbound[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
-qed
+qed
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
using real_arch_pow[of 2 x] by simp
@@ -2777,13 +2777,13 @@
lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
shows "\<exists>n. x^n < y"
proof-
- {assume x0: "x > 0"
+ {assume x0: "x > 0"
from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
from real_arch_pow[OF ix, of "1/y"]
obtain n where n: "1/y < (1/x)^n" by blast
- then
+ then
have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
- moreover
+ moreover
{assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
ultimately show ?thesis by metis
qed
@@ -2821,18 +2821,18 @@
have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
by (simp add: linorder_linear)
ultimately show ?thesis by arith
-qed
+qed
lemma real_min_rinf: "min x y = rinf {x,y}"
proof-
have f: "finite {x, y}" "{x,y} \<noteq> {}" by simp_all
- from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
+ from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
by (simp add: linorder_linear)
moreover
have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
by simp
ultimately show ?thesis by arith
-qed
+qed
(* ------------------------------------------------------------------------- *)
(* Geometric progression. *)
@@ -2863,9 +2863,9 @@
from mn have mn': "n - m \<ge> 0" by arith
let ?f = "op + m"
have i: "inj_on ?f ?S" unfolding inj_on_def by auto
- have f: "?f ` ?S = {m..n}"
+ have f: "?f ` ?S = {m..n}"
using mn apply (auto simp add: image_iff Bex_def) by arith
- have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
+ have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
by (rule ext, simp add: power_add power_mult)
from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
@@ -2873,8 +2873,8 @@
by (simp add: ring_simps power_add[symmetric])
qed
-lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} =
- (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
+lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} =
+ (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
else (x^ m - x^ (Suc n)) / (1 - x))"
proof-
{assume nm: "n < m" hence ?thesis by simp}
@@ -2889,7 +2889,7 @@
ultimately show ?thesis by metis
qed
-lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} =
+lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
unfolding sum_gp[of x m "m + n"] power_Suc
by (simp add: ring_simps power_add)
@@ -2908,7 +2908,7 @@
lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
by (metis subspace_def)
lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
@@ -2926,10 +2926,10 @@
shows "setsum f B \<in> A"
using fB f sA
apply(induct rule: finite_induct[OF fB])
- by (simp add: subspace_def sA, auto simp add: sA subspace_add)
-
-lemma subspace_linear_image:
- assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
+ by (simp add: subspace_def sA, auto simp add: sA subspace_add)
+
+lemma subspace_linear_image:
+ assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
shows "subspace(f ` S)"
using lf sS linear_0[OF lf]
unfolding linear_def subspace_def
@@ -2986,7 +2986,7 @@
from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
from P have P': "P \<in> subspace" by (simp add: mem_def)
from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
- show "P x" by (metis mem_def subset_eq)
+ show "P x" by (metis mem_def subset_eq)
qed
lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
@@ -3016,11 +3016,11 @@
using span_induct SP P by blast
inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
- where
+ where
span_induct_alt_help_0: "span_induct_alt_help S 0"
| span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *s x + z)"
-lemma span_induct_alt':
+lemma span_induct_alt':
assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" shows "\<forall>x \<in> span S. h x"
proof-
{fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
@@ -3031,7 +3031,7 @@
done}
note th0 = this
{fix x assume x: "x \<in> span S"
-
+
have "span_induct_alt_help S x"
proof(rule span_induct[where x=x and S=S])
show "x \<in> span S" using x .
@@ -3043,7 +3043,7 @@
have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
moreover
{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
- from h
+ from h
have "span_induct_alt_help S (x + y)"
apply (induct rule: span_induct_alt_help.induct)
apply simp
@@ -3054,7 +3054,7 @@
done}
moreover
{fix c x assume xt: "span_induct_alt_help S x"
- then have "span_induct_alt_help S (c*s x)"
+ then have "span_induct_alt_help S (c*s x)"
apply (induct rule: span_induct_alt_help.induct)
apply (simp add: span_induct_alt_help_0)
apply (simp add: vector_smult_assoc vector_add_ldistrib)
@@ -3063,13 +3063,13 @@
apply simp
done
}
- ultimately show "subspace (span_induct_alt_help S)"
+ ultimately show "subspace (span_induct_alt_help S)"
unfolding subspace_def mem_def Ball_def by blast
qed}
with th0 show ?thesis by blast
-qed
-
-lemma span_induct_alt:
+qed
+
+lemma span_induct_alt:
assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" and x: "x \<in> span S"
shows "h x"
using span_induct_alt'[of h S] h0 hS x by blast
@@ -3118,9 +3118,9 @@
apply (rule subspace_span)
apply (rule x)
done}
- moreover
+ moreover
{fix x assume x: "x \<in> span S"
- have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
+ have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
unfolding mem_def Collect_def ..
have "f x \<in> span (f ` S)"
apply (rule span_induct[where S=S])
@@ -3146,15 +3146,15 @@
apply (rule exI[where x="1"], simp)
by (rule span_0)}
moreover
- {assume ab: "x \<noteq> b"
+ {assume ab: "x \<noteq> b"
then have "?P x" using xS
apply -
apply (rule exI[where x=0])
apply (rule span_superset)
by simp}
ultimately have "?P x" by blast}
- moreover have "subspace ?P"
- unfolding subspace_def
+ moreover have "subspace ?P"
+ unfolding subspace_def
apply auto
apply (simp add: mem_def)
apply (rule exI[where x=0])
@@ -3174,7 +3174,7 @@
apply (rule span_mul[unfolded mem_def])
apply assumption
by (vector ring_simps)
- ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
+ ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
qed
lemma span_breakdown_eq:
@@ -3186,7 +3186,7 @@
apply (rule_tac x= "k" in exI)
apply (rule set_rev_mp[of _ "span (S - {a})" _])
apply assumption
- apply (rule span_mono)
+ apply (rule span_mono)
apply blast
done}
moreover
@@ -3196,7 +3196,7 @@
apply (rule span_add)
apply (rule set_rev_mp[of _ "span S" _])
apply (rule k)
- apply (rule span_mono)
+ apply (rule span_mono)
apply blast
apply (rule span_mul)
apply (rule span_superset)
@@ -3224,7 +3224,7 @@
done
with na have ?thesis by blast}
moreover
- {assume k0: "k \<noteq> 0"
+ {assume k0: "k \<noteq> 0"
have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
by (vector field_simps)
@@ -3247,8 +3247,8 @@
ultimately show ?thesis by blast
qed
-lemma in_span_delete:
- assumes a: "(a::'a::field^'n) \<in> span S"
+lemma in_span_delete:
+ assumes a: "(a::'a::field^'n) \<in> span S"
and na: "a \<notin> span (S-{b})"
shows "b \<in> span (insert a (S - {b}))"
apply (rule in_span_insert)
@@ -3268,7 +3268,7 @@
from span_breakdown[of x "insert x S" y, OF insertI1 y]
obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
have eq: "y = (y - k *s x) + k *s x" by vector
- show ?thesis
+ show ?thesis
apply (subst eq)
apply (rule span_add)
apply (rule set_rev_mp)
@@ -3304,18 +3304,18 @@
next
fix c x y
assume x: "x \<in> P" and hy: "?h y"
- from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
+ from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
let ?S = "insert x S"
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
else u y"
from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
{assume xS: "x \<in> S"
- have S1: "S = (S - {x}) \<union> {x}"
+ have S1: "S = (S - {x}) \<union> {x}"
and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
- using xS
- by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
+ using xS
+ by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
setsum_clauses(2)[OF fS] cong del: if_weak_cong)
also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
@@ -3324,7 +3324,7 @@
by (simp add: add_commute u)
finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
- moreover
+ moreover
{assume xS: "x \<notin> S"
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
unfolding u[symmetric]
@@ -3334,7 +3334,7 @@
by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
ultimately have "?Q ?S ?u (c*s x + y)"
by (cases "x \<in> S", simp, simp)
- then show "?h (c*s x + y)"
+ then show "?h (c*s x + y)"
apply -
apply (rule exI[where x="?S"])
apply (rule exI[where x="?u"]) by metis
@@ -3346,11 +3346,11 @@
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>(v::'a::{idom,field}^'n) \<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *s v) S = 0))" (is "?lhs = ?rhs")
proof-
{assume dP: "dependent P"
- then obtain a S u where aP: "a \<in> P" and fS: "finite S"
- and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
+ then obtain a S u where aP: "a \<in> P" and fS: "finite S"
+ and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
unfolding dependent_def span_explicit by blast
- let ?S = "insert a S"
- let ?u = "\<lambda>y. if y = a then - 1 else u y"
+ let ?S = "insert a S"
+ let ?u = "\<lambda>y. if y = a then - 1 else u y"
let ?v = a
from aP SP have aS: "a \<notin> S" by blast
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
@@ -3366,16 +3366,16 @@
apply (rule exI[where x= "?u"])
by clarsimp}
moreover
- {fix S u v assume fS: "finite S"
- and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
+ {fix S u v assume fS: "finite S"
+ and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
and u: "setsum (\<lambda>v. u v *s v) S = 0"
- let ?a = v
+ let ?a = v
let ?S = "S - {v}"
let ?u = "\<lambda>i. (- u i) / u v"
- have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" using fS SP vS by auto
+ have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" using fS SP vS by auto
have "setsum (\<lambda>v. ?u v *s v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v"
- using fS vS uv
- by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
+ using fS vS uv
+ by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
vector_smult_assoc field_simps)
also have "\<dots> = ?a"
unfolding setsum_cmul u
@@ -3398,7 +3398,7 @@
(is "_ = ?rhs")
proof-
{fix y assume y: "y \<in> span S"
- from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
+ from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
@@ -3410,7 +3410,7 @@
done
hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
hence "y \<in> ?rhs" by auto}
- moreover
+ moreover
{fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
then have "y \<in> span S" using fS unfolding span_explicit by auto}
ultimately show ?thesis by blast
@@ -3431,7 +3431,7 @@
apply (auto simp add: Collect_def mem_def)
done
-
+
lemma has_size_stdbasis: "{basis i ::real ^'n | i. i \<in> {1 .. dimindex (UNIV :: 'n set)}} hassize (dimindex(UNIV :: 'n set))" (is "?S hassize ?n")
proof-
have eq: "?S = basis ` {1 .. ?n}" by blast
@@ -3461,10 +3461,10 @@
{fix x::"'a^'n" assume xS: "x\<in> ?B"
from xS have "?P x" by (auto simp add: basis_component)}
moreover
- have "subspace ?P"
+ have "subspace ?P"
by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component)
ultimately show ?thesis
- using x span_induct[of ?B ?P x] i iS by blast
+ using x span_induct[of ?B ?P x] i iS by blast
qed
lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
@@ -3508,7 +3508,7 @@
apply assumption
apply blast
by (simp add: dependent_def)}
- moreover
+ moreover
{assume i: ?rhs
have ?lhs using i aS
apply simp
@@ -3541,7 +3541,7 @@
by (metis subset_eq span_superset)
lemma spanning_subset_independent:
- assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
+ assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
and AsB: "A \<subseteq> span B"
shows "A = B"
proof
@@ -3569,7 +3569,7 @@
lemma exchange_lemma:
assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
- and sp:"s \<subseteq> span t"
+ and sp:"s \<subseteq> span t"
shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
@@ -3584,15 +3584,15 @@
and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
and n: "n = card (t - s)"
let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
- let ?ths = "\<exists>t'. ?P t'"
- {assume st: "s \<subseteq> t"
- from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+ let ?ths = "\<exists>t'. ?P t'"
+ {assume st: "s \<subseteq> t"
+ from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto simp add: hassize_def intro: span_superset)}
moreover
{assume st: "t \<subseteq> s"
-
- from spanning_subset_independent[OF st s sp]
- st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+
+ from spanning_subset_independent[OF st s sp]
+ st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto simp add: hassize_def intro: span_superset)}
moreover
{assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
@@ -3603,28 +3603,28 @@
from b ft have ct0: "card t \<noteq> 0" by auto
{assume stb: "s \<subseteq> span(t -{b})"
from ft have ftb: "finite (t -{b})" by auto
- from H[rule_format, OF cardlt ftb s stb]
+ from H[rule_format, OF cardlt ftb s stb]
obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
let ?w = "insert b u"
have th0: "s \<subseteq> insert b u" using u by blast
- from u(3) b have "u \<subseteq> s \<union> t" by blast
+ from u(3) b have "u \<subseteq> s \<union> t" by blast
then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
have bu: "b \<notin> u" using b u by blast
from u(1) have fu: "finite u" by (simp add: hassize_def)
from u(1) ft b have "u hassize (card t - 1)" by auto
- then
- have th2: "insert b u hassize card t"
+ then
+ have th2: "insert b u hassize card t"
using card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
from u(4) have "s \<subseteq> span u" .
also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
finally have th3: "s \<subseteq> span (insert b u)" . from th0 th1 th2 th3 have th: "?P ?w" by blast
from th have ?ths by blast}
moreover
- {assume stb: "\<not> s \<subseteq> span(t -{b})"
+ {assume stb: "\<not> s \<subseteq> span(t -{b})"
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
have ab: "a \<noteq> b" using a b by blast
have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
- have mlt: "card ((insert a (t - {b})) - s) < n"
+ have mlt: "card ((insert a (t - {b})) - s) < n"
using cardlt ft n a b by auto
have ft': "finite (insert a (t - {b}))" using ft by auto
{fix x assume xs: "x \<in> s"
@@ -3637,15 +3637,15 @@
have x: "x \<in> span (insert b (insert a (t - {b})))" ..
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .}
then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
-
- from H[rule_format, OF mlt ft' s sp' refl] obtain u where
+
+ from H[rule_format, OF mlt ft' s sp' refl] obtain u where
u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
"s \<subseteq> span u" by blast
from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
then have ?ths by blast }
ultimately have ?ths by blast
}
- ultimately
+ ultimately
show ?ths by blast
qed
@@ -3659,7 +3659,7 @@
lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
proof-
have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
- show ?thesis unfolding eq
+ show ?thesis unfolding eq
apply (rule finite_imageI)
apply (rule finite_intvl)
done
@@ -3668,7 +3668,7 @@
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> {(i::nat) .. j}}"
proof-
have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
- show ?thesis unfolding eq
+ show ?thesis unfolding eq
apply (rule finite_imageI)
apply (rule finite_atLeastAtMost)
done
@@ -3682,7 +3682,7 @@
apply (rule independent_span_bound)
apply (rule finite_Atleast_Atmost_nat)
apply assumption
- unfolding span_stdbasis
+ unfolding span_stdbasis
apply (rule subset_UNIV)
done
@@ -3710,14 +3710,14 @@
from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
from a have aS: "a \<notin> S" by (auto simp add: span_superset)
have th0: "insert a S \<subseteq> V" using a sv by blast
- from independent_insert[of a S] i a
+ from independent_insert[of a S] i a
have th1: "independent (insert a S)" by auto
- have mlt: "?d - card (insert a S) < n"
- using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"]
- by auto
-
- from H[rule_format, OF mlt th0 th1 refl]
- obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
+ have mlt: "?d - card (insert a S) < n"
+ using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"]
+ by auto
+
+ from H[rule_format, OF mlt th0 th1 refl]
+ obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto
then have ?ths by blast}
@@ -3732,7 +3732,7 @@
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
-lemma basis_exists: "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
+lemma basis_exists: "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n)"]
unfolding hassize_def
using maximal_independent_subset[of V] independent_bound
@@ -3784,7 +3784,7 @@
qed
lemma card_le_dim_spanning:
- assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B"
+ assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B"
and fB: "finite B" and dVB: "dim V \<ge> card B"
shows "independent B"
proof-
@@ -3794,10 +3794,10 @@
from BV a have th0: "B -{a} \<subseteq> V" by blast
{fix x assume x: "x \<in> V"
from a have eq: "insert a (B -{a}) = B" by blast
- from x VB have x': "x \<in> span B" by blast
+ from x VB have x': "x \<in> span B" by blast
from span_trans[OF a(2), unfolded eq, OF x']
have "x \<in> span (B -{a})" . }
- then have th1: "V \<subseteq> span (B -{a})" by blast
+ then have th1: "V \<subseteq> span (B -{a})" by blast
have th2: "finite (B -{a})" using fB by auto
from span_card_ge_dim[OF th0 th1 th2]
have c: "dim V \<le> card (B -{a})" .
@@ -3806,7 +3806,7 @@
qed
lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
- by (metis hassize_def order_eq_iff card_le_dim_spanning
+ by (metis hassize_def order_eq_iff card_le_dim_spanning
card_ge_dim_independent)
(* ------------------------------------------------------------------------- *)
@@ -3818,18 +3818,18 @@
by (metis independent_card_le_dim independent_bound subset_refl)
lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
- using independent_bound_general[of S] by (metis linorder_not_le)
+ using independent_bound_general[of S] by (metis linorder_not_le)
lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
proof-
- have th0: "dim S \<le> dim (span S)"
+ have th0: "dim S \<le> dim (span S)"
by (auto simp add: subset_eq intro: dim_subset span_superset)
- from basis_exists[of S]
+ from basis_exists[of S]
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
- have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
- have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
- from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
+ have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
+ have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
+ from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
using fB(2) by arith
qed
@@ -3847,7 +3847,7 @@
lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
proof-
- from basis_exists[of S] obtain B where
+ from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
have "dim (f ` S) \<le> card (f ` B)"
@@ -3860,7 +3860,7 @@
(* Relation between bases and injectivity/surjectivity of map. *)
lemma spanning_surjective_image:
- assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
+ assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
and lf: "linear f" and sf: "surj f"
shows "UNIV \<subseteq> span (f ` S)"
proof-
@@ -3881,7 +3881,7 @@
hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
with a(1) iS have False by (simp add: dependent_def) }
then show ?thesis unfolding dependent_def by blast
-qed
+qed
(* ------------------------------------------------------------------------- *)
(* Picking an orthogonal replacement for a spanning set. *)
@@ -3904,15 +3904,15 @@
case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
next
case (2 a B)
- note fB = `finite B` and aB = `a \<notin> B`
- from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
- obtain C where C: "finite C" "card C \<le> card B"
+ note fB = `finite B` and aB = `a \<notin> B`
+ from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
+ obtain C where C: "finite C" "card C \<le> card B"
"span C = span B" "pairwise orthogonal C" by blast
let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
let ?C = "insert ?a C"
from C(1) have fC: "finite ?C" by simp
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
- {fix x k
+ {fix x k
have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
have "x - k *s (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *s x)) \<in> span C \<longleftrightarrow> x - k *s a \<in> span C"
apply (simp only: vector_ssub_ldistrib th0)
@@ -3924,18 +3924,18 @@
by (rule span_superset)}
then have SC: "span ?C = span (insert a B)"
unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
- thm pairwise_def
+ thm pairwise_def
{fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
- {assume xa: "x = ?a" and ya: "y = ?a"
+ {assume xa: "x = ?a" and ya: "y = ?a"
have "orthogonal x y" using xa ya xy by blast}
moreover
- {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
+ {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
from ya have Cy: "C = insert y (C - {y})" by blast
have fth: "finite (C - {y})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
- apply simp
+ apply simp
apply (subst Cy)
using C(1) fth
apply (simp only: setsum_clauses)
@@ -3946,13 +3946,13 @@
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
- {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
+ {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
from xa have Cx: "C = insert x (C - {x})" by blast
have fth: "finite (C - {x})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
- apply simp
+ apply simp
apply (subst Cx)
using C(1) fth
apply (simp only: setsum_clauses)
@@ -3963,12 +3963,12 @@
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
- {assume xa: "x \<in> C" and ya: "y \<in> C"
+ {assume xa: "x \<in> C" and ya: "y \<in> C"
have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
ultimately have "orthogonal x y" using xC yC by blast}
then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
from fC cC SC CPO have "?P (insert a B) ?C" by blast
- then show ?case by blast
+ then show ?case by blast
qed
lemma orthogonal_basis_exists:
@@ -3977,18 +3977,18 @@
proof-
from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
- from basis_orthogonal[OF fB(1)] obtain C where
+ from basis_orthogonal[OF fB(1)] obtain C where
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
- from C B
- have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
+ from C B
+ have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
- have iC: "independent C" by (simp add: dim_span)
+ have iC: "independent C" by (simp add: dim_span)
from C fB have "card C \<le> dim V" by simp
moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
by (simp add: dim_span)
ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
- from C B CSV CdV iC show ?thesis by auto
+ from C B CSV CdV iC show ?thesis by auto
qed
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
@@ -4003,8 +4003,8 @@
shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
proof-
from sU obtain a where a: "a \<notin> span S" by blast
- from orthogonal_basis_exists obtain B where
- B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
+ from orthogonal_basis_exists obtain B where
+ B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
by blast
from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
from span_mono[OF B(2)] span_mono[OF B(3)]
@@ -4020,12 +4020,12 @@
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
proof(rule span_induct')
show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
- by (auto simp add: subspace_def mem_def dot_radd dot_rmult)
+ by (auto simp add: subspace_def mem_def dot_radd dot_rmult)
next
{fix x assume x: "x \<in> B"
from x have B': "B = insert x (B - {x})" by blast
have fth: "finite (B - {x})" using fB by simp
- have "?a \<bullet> x = 0"
+ have "?a \<bullet> x = 0"
apply (subst B') using fB fth
unfolding setsum_clauses(2)[OF fth]
apply simp
@@ -4038,7 +4038,7 @@
with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
qed
-lemma span_not_univ_subset_hyperplane:
+lemma span_not_univ_subset_hyperplane:
assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
using span_not_univ_orthogonal[OF SU] by auto
@@ -4058,9 +4058,9 @@
(* We can extend a linear basis-basis injection to the whole set. *)
lemma linear_indep_image_lemma:
- assumes lf: "linear f" and fB: "finite B"
+ assumes lf: "linear f" and fB: "finite B"
and ifB: "independent (f ` B)"
- and fi: "inj_on f B" and xsB: "x \<in> span B"
+ and fi: "inj_on f B" and xsB: "x \<in> span B"
and fx: "f (x::'a::field^'n) = 0"
shows "x = 0"
using fB ifB fi xsB fx
@@ -4070,11 +4070,11 @@
case (2 a b x)
have fb: "finite b" using "2.prems" by simp
have th0: "f ` b \<subseteq> f ` (insert a b)"
- apply (rule image_mono) by blast
+ apply (rule image_mono) by blast
from independent_mono[ OF "2.prems"(2) th0]
have ifb: "independent (f ` b)" .
- have fib: "inj_on f b"
- apply (rule subset_inj_on [OF "2.prems"(3)])
+ have fib: "inj_on f b"
+ apply (rule subset_inj_on [OF "2.prems"(3)])
by blast
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
@@ -4084,16 +4084,16 @@
using k span_mono[of "b-{a}" b] by blast
hence "f x - k*s f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
- hence th: "-k *s f a \<in> span (f ` b)"
+ hence th: "-k *s f a \<in> span (f ` b)"
using "2.prems"(5) by (simp add: vector_smult_lneg)
- {assume k0: "k = 0"
+ {assume k0: "k = 0"
from k0 k have "x \<in> span (b -{a})" by simp
then have "x \<in> span b" using span_mono[of "b-{a}" b]
by blast}
moreover
{assume k0: "k \<noteq> 0"
from span_mul[OF th, of "- 1/ k"] k0
- have th1: "f a \<in> span (f ` b)"
+ have th1: "f a \<in> span (f ` b)"
by (auto simp add: vector_smult_assoc)
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
@@ -4112,17 +4112,17 @@
lemma linear_independent_extend_lemma:
assumes fi: "finite B" and ib: "independent B"
- shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
+ shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
\<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
\<and> (\<forall>x\<in> B. g x = f x)"
using ib fi
proof(induct rule: finite_induct[OF fi])
- case 1 thus ?case by (auto simp add: span_empty)
+ case 1 thus ?case by (auto simp add: span_empty)
next
case (2 a b)
from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
by (simp_all add: independent_insert)
- from "2.hyps"(3)[OF ibf] obtain g where
+ from "2.hyps"(3)[OF ibf] obtain g where
g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
"\<forall>x\<in>span b. \<forall>c. g (c *s x) = c *s g x" "\<forall>x\<in>b. g x = f x" by blast
let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
@@ -4132,12 +4132,12 @@
unfolding span_breakdown_eq[symmetric]
using z .
{fix k assume k: "z - k *s a \<in> span b"
- have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
+ have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
from span_sub[OF th0 k]
have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
{assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
- from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
+ from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> span b" by (simp add: vector_smult_assoc)
with "2.prems"(1) "2.hyps"(2) have False
by (auto simp add: dependent_def)}
@@ -4146,26 +4146,26 @@
note h = this
let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
{fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
- have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
+ have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
by (vector ring_simps)
have addh: "?h (x + y) = ?h x + ?h y"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (rule span_add[OF x y])
unfolding tha
by (metis span_add x y conjunct1[OF h, rule_format])
- have "?g (x + y) = ?g x + ?g y"
+ have "?g (x + y) = ?g x + ?g y"
unfolding addh tha
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: vector_sadd_rdistrib)}
moreover
{fix x:: "'a^'n" and c:: 'a assume x: "x \<in> span (insert a b)"
- have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
+ have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
by (vector ring_simps)
- have hc: "?h (c *s x) = c * ?h x"
+ have hc: "?h (c *s x) = c * ?h x"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (metis span_mul x)
by (metis tha span_mul x conjunct1[OF h])
- have "?g (c *s x) = c*s ?g x"
+ have "?g (c *s x) = c*s ?g x"
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
by (vector ring_simps)}
moreover
@@ -4177,7 +4177,7 @@
using conjunct1[OF h, OF span_superset, OF insertI1]
by (auto simp add: span_0)
- from xa ha1[symmetric] have "?g x = f x"
+ from xa ha1[symmetric] have "?g x = f x"
apply simp
using g(2)[rule_format, OF span_0, of 0]
by simp}
@@ -4201,12 +4201,12 @@
proof-
from maximal_independent_subset_extend[of B UNIV] iB
obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
-
+
from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
- obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
+ obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
\<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
\<and> (\<forall>x\<in> C. g x = f x)" by blast
- from g show ?thesis unfolding linear_def using C
+ from g show ?thesis unfolding linear_def using C
apply clarsimp by blast
qed
@@ -4218,7 +4218,7 @@
proof(induct arbitrary: B rule: finite_induct[OF fA])
case 1 thus ?case by simp
next
- case (2 x s t)
+ case (2 x s t)
thus ?case
proof(induct rule: finite_induct[OF "2.prems"(1)])
case 1 then show ?case by simp
@@ -4234,7 +4234,7 @@
qed
qed
-lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
+lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
c: "card A = card B"
shows "A = B"
proof-
@@ -4245,27 +4245,27 @@
from card_Un_disjoint[OF fA fBA e, unfolded eq c]
have "card (B - A) = 0" by arith
hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
- with AB show "A = B" by blast
+ with AB show "A = B" by blast
qed
lemma subspace_isomorphism:
- assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T"
+ assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T"
and d: "dim S = dim T"
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
proof-
- from basis_exists[of S] obtain B where
+ from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
- from basis_exists[of T] obtain C where
+ from basis_exists[of T] obtain C where
C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
from B(4) C(4) card_le_inj[of B C] d obtain f where
- f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
+ f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
from linear_independent_extend[OF B(2)] obtain g where
g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
from B(4) have fB: "finite B" by (simp add: hassize_def)
from C(4) have fC: "finite C" by (simp add: hassize_def)
- from inj_on_iff_eq_card[OF fB, of f] f(2)
+ from inj_on_iff_eq_card[OF fB, of f] f(2)
have "card (f ` B) = card B" by simp
- with B(4) C(4) have ceq: "card (f ` B) = card C" using d
+ with B(4) C(4) have ceq: "card (f ` B) = card C" using d
by (simp add: hassize_def)
have "g ` B = f ` B" using g(2)
by (auto simp add: image_iff)
@@ -4277,9 +4277,9 @@
{fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
- have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
+ have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
have "x=y" using g0[OF th1 th0] by simp }
- then have giS: "inj_on g S"
+ then have giS: "inj_on g S"
unfolding inj_on_def by blast
from span_subspace[OF B(1,3) s]
have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
@@ -4308,20 +4308,20 @@
qed
lemma linear_eq_0:
- assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
+ assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
lemma linear_eq:
assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
- and fg: "\<forall> x\<in> B. f x = g x"
+ and fg: "\<forall> x\<in> B. f x = g x"
shows "\<forall>x\<in> S. f x = g x"
proof-
let ?h = "\<lambda>x. f x - g x"
from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
show ?thesis by simp
-qed
+qed
lemma linear_eq_stdbasis:
assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
@@ -4329,7 +4329,7 @@
shows "f = g"
proof-
let ?U = "UNIV :: 'm set"
- let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}"
+ let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}"
{fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
from equalityD2[OF span_stdbasis]
have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
@@ -4341,27 +4341,27 @@
(* Similar results for bilinear functions. *)
lemma bilinear_eq:
- assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
+ assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
and bg: "bilinear g"
and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
proof-
let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
- from bf bg have sp: "subspace ?P"
- unfolding bilinear_def linear_def subspace_def bf bg
+ from bf bg have sp: "subspace ?P"
+ unfolding bilinear_def linear_def subspace_def bf bg
by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf])
- have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
+ have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
apply -
apply (rule ballI)
- apply (rule span_induct[of B ?P])
+ apply (rule span_induct[of B ?P])
defer
apply (rule sp)
apply assumption
apply (clarsimp simp add: Ball_def)
apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
- using fg
+ using fg
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_def
by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf])
@@ -4369,7 +4369,7 @@
qed
lemma bilinear_eq_stdbasis:
- assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
+ assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
and bg: "bilinear g"
and fg: "\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. \<forall>j\<in> {1 .. dimindex (UNIV :: 'n set)}. f (basis i) (basis j) = g (basis i) (basis j)"
shows "f = g"
@@ -4394,16 +4394,16 @@
proof-
from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> {1 .. dimindex (UNIV::'n set)}}. h x = inv f x" by blast
- from h(2)
+ from h(2)
have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (h \<circ> f) (basis i) = id (basis i)"
using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
apply auto
apply (erule_tac x="basis i" in allE)
by auto
-
+
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
have "h o f = id" .
- then show ?thesis using h(1) by blast
+ then show ?thesis using h(1) by blast
qed
lemma linear_surjective_right_inverse:
@@ -4411,18 +4411,18 @@
shows "\<exists>g. linear g \<and> f o g = id"
proof-
from linear_independent_extend[OF independent_stdbasis]
- obtain h:: "real ^'n \<Rightarrow> real ^'m" where
+ obtain h:: "real ^'n \<Rightarrow> real ^'m" where
h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast
- from h(2)
+ from h(2)
have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)"
using sf
apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
apply (erule_tac x="basis i" in allE)
by auto
-
+
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
have "f o h = id" .
- then show ?thesis using h(1) by blast
+ then show ?thesis using h(1) by blast
qed
lemma matrix_left_invertible_injective:
@@ -4434,7 +4434,7 @@
unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
moreover
{assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
- hence i: "inj (op *v A)" unfolding inj_on_def by auto
+ hence i: "inj (op *v A)" unfolding inj_on_def by auto
from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
obtain g where g: "linear g" "g o op *v A = id" by blast
have "matrix g ** A = mat 1"
@@ -4454,25 +4454,25 @@
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
proof-
{fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1"
- {fix x :: "real ^ 'm"
+ {fix x :: "real ^ 'm"
have "A *v (B *v x) = x"
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
hence "surj (op *v A)" unfolding surj_def by metis }
moreover
{assume sf: "surj (op *v A)"
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
- obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
+ obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
by blast
have "A ** (matrix g) = mat 1"
- unfolding matrix_eq matrix_vector_mul_lid
- matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
+ unfolding matrix_eq matrix_vector_mul_lid
+ matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) unfolding o_def stupid_ext[symmetric] id_def
.
hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
}
ultimately show ?thesis unfolding surj_def by blast
-qed
+qed
lemma matrix_left_invertible_independent_columns:
fixes A :: "real^'n^'m"
@@ -4481,7 +4481,7 @@
proof-
let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
{assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
- {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
+ {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
and i: "i \<in> ?U"
let ?x = "\<chi> i. c i"
have th0:"A *v ?x = 0"
@@ -4493,11 +4493,11 @@
hence ?rhs by blast}
moreover
{assume H: ?rhs
- {fix x assume x: "A *v x = 0"
+ {fix x assume x: "A *v x = 0"
let ?c = "\<lambda>i. ((x$i ):: real)"
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
have "x = 0" by vector}}
- ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
+ ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
qed
lemma matrix_right_invertible_independent_rows:
@@ -4514,13 +4514,13 @@
have fU: "finite ?U" by simp
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
- apply (subst eq_commute) ..
+ apply (subst eq_commute) ..
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
{assume h: ?lhs
- {fix x:: "real ^'n"
+ {fix x:: "real ^'n"
from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
- have "x \<in> span (columns A)"
+ have "x \<in> span (columns A)"
unfolding y[symmetric]
apply (rule span_setsum[OF fU])
apply clarify
@@ -4532,21 +4532,21 @@
moreover
{assume h:?rhs
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
- {fix y have "?P y"
+ {fix y have "?P y"
proof(rule span_induct_alt[of ?P "columns A"])
show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
apply (rule exI[where x=0])
by (simp add: zero_index vector_smult_lzero)
next
fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
- from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
+ from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
unfolding columns_def by blast
- from y2 obtain x:: "real ^'m" where
+ from y2 obtain x:: "real ^'m" where
x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
show "?P (c*s y1 + y2)"
proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric])
- fix j
+ fix j
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)
by (simp add: ring_simps)
@@ -4558,7 +4558,7 @@
by (simp add: setsum_addf)
also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
unfolding setsum_delta[OF fU]
- using i(1) by simp
+ using i(1) by simp
finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
qed
@@ -4579,12 +4579,12 @@
(* An injective map real^'n->real^'n is also surjective. *)
lemma linear_injective_imp_surjective:
- assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f"
+ assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f"
shows "surj f"
proof-
let ?U = "UNIV :: (real ^'n) set"
- from basis_exists[of ?U] obtain B
- where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
+ from basis_exists[of ?U] obtain B
+ where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
by blast
from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
have th: "?U \<subseteq> span (f ` B)"
@@ -4604,7 +4604,7 @@
(* And vice versa. *)
-lemma surjective_iff_injective_gen:
+lemma surjective_iff_injective_gen:
assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
and ST: "f ` S \<subseteq> T"
shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
@@ -4641,17 +4641,17 @@
qed
lemma linear_surjective_imp_injective:
- assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f"
+ assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f"
shows "inj f"
proof-
let ?U = "UNIV :: (real ^'n) set"
- from basis_exists[of ?U] obtain B
- where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
+ from basis_exists[of ?U] obtain B
+ where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
by blast
{fix x assume x: "x \<in> span B" and fx: "f x = 0"
from B(4) have fB: "finite B" by (simp add: hassize_def)
from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
- have fBi: "independent (f ` B)"
+ have fBi: "independent (f ` B)"
apply (rule card_le_dim_spanning[of "f ` B" ?U])
apply blast
using sf B(3)
@@ -4676,12 +4676,12 @@
moreover have "card (f ` B) \<le> card B"
by (rule card_image_le, rule fB)
ultimately have th1: "card B = card (f ` B)" unfolding d by arith
- have fiB: "inj_on f B"
+ have fiB: "inj_on f B"
unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
have "x = 0" by blast}
note th = this
- from th show ?thesis unfolding linear_injective_0[OF lf]
+ from th show ?thesis unfolding linear_injective_0[OF lf]
using B(3) by blast
qed
@@ -4689,7 +4689,7 @@
lemma left_right_inverse_eq:
assumes fg: "f o g = id" and gh: "g o h = id"
- shows "f = h"
+ shows "f = h"
proof-
have "f = f o (g o h)" unfolding gh by simp
also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
@@ -4723,7 +4723,7 @@
{fix f f':: "real ^'n \<Rightarrow> real ^'n"
assume lf: "linear f" "linear f'" and f: "f o f' = id"
from f have sf: "surj f"
-
+
apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
by metis
from linear_surjective_isomorphism[OF lf(1) sf] lf f
@@ -4735,13 +4735,13 @@
(* Moreover, a one-sided inverse is automatically linear. *)
lemma left_inverse_linear:
- assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id"
+ assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id"
shows "linear g"
proof-
from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
by metis
- from linear_injective_isomorphism[OF lf fi]
- obtain h:: "real ^'n \<Rightarrow> real ^'n" where
+ from linear_injective_isomorphism[OF lf fi]
+ obtain h:: "real ^'n \<Rightarrow> real ^'n" where
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
have "h = g" apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def stupid_ext[symmetric])
@@ -4750,13 +4750,13 @@
qed
lemma right_inverse_linear:
- assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id"
+ assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id"
shows "linear g"
proof-
from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
by metis
- from linear_surjective_isomorphism[OF lf fi]
- obtain h:: "real ^'n \<Rightarrow> real ^'n" where
+ from linear_surjective_isomorphism[OF lf fi]
+ obtain h:: "real ^'n \<Rightarrow> real ^'n" where
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
have "h = g" apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def stupid_ext[symmetric])
@@ -4767,7 +4767,7 @@
(* The same result in terms of square matrices. *)
lemma matrix_left_right_inverse:
- fixes A A' :: "real ^'n^'n"
+ fixes A A' :: "real ^'n^'n"
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
proof-
{fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
@@ -4779,7 +4779,7 @@
from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
obtain f' :: "real ^'n \<Rightarrow> real ^'n"
where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
- have th: "matrix f' ** A = mat 1"
+ have th: "matrix f' ** A = mat 1"
by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
@@ -4846,17 +4846,17 @@
have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
- show ?thesis
+ show ?thesis
unfolding infnorm_def
unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
apply (subst diff_le_eq[symmetric])
unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
- unfolding infnorm_set_image bex_simps
+ unfolding infnorm_set_image bex_simps
apply (subst th)
- unfolding th1
+ unfolding th1
unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
-
- unfolding infnorm_set_image ball_simps bex_simps
+
+ unfolding infnorm_set_image ball_simps bex_simps
apply (simp add: vector_add_component)
apply (metis numseg_dimindex_nonempty th2)
done
@@ -4885,7 +4885,7 @@
apply (simp add: vector_component)
done
-lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
+lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
proof-
have "y - x = - (x - y)" by simp
then show ?thesis by (metis infnorm_neg)
@@ -4896,7 +4896,7 @@
have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
- have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
+ have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
"infnorm y \<le> infnorm (x - y) + infnorm x"
by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
from th[OF ths] show ?thesis .
@@ -4911,11 +4911,11 @@
let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
have fS: "finite ?S" unfolding image_Collect[symmetric]
- apply (rule finite_imageI) unfolding Collect_def mem_def by simp
+ apply (rule finite_imageI) unfolding Collect_def mem_def by simp
have S0: "?S \<noteq> {}" using numseg_dimindex_nonempty by blast
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i
- show ?thesis unfolding infnorm_def isUb_def setle_def
+ show ?thesis unfolding infnorm_def isUb_def setle_def
unfolding infnorm_set_image ball_simps by auto
qed
@@ -4942,7 +4942,7 @@
have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
unfolding th by simp
with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *s x))" by (simp add: field_simps)
- then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
+ then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
using ap by (simp add: field_simps)
with infnorm_mul_lemma[of a x] have ?thesis by arith }
ultimately show ?thesis by blast
@@ -4954,7 +4954,7 @@
(* Prove that it differs only up to a bound from Euclidean norm. *)
lemma infnorm_le_norm: "infnorm x \<le> norm x"
- unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
+ unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
by (metis component_le_norm)
lemma card_enum: "card {1 .. n} = n" by auto
@@ -4968,20 +4968,20 @@
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
- unfolding power_mult_distrib d2
+ unfolding power_mult_distrib d2
apply (subst d)
apply (subst power2_abs[symmetric])
unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
apply (subst power2_abs[symmetric])
apply (rule setsum_bounded)
apply (rule power_mono)
- unfolding abs_of_nonneg[OF infnorm_pos_le]
+ unfolding abs_of_nonneg[OF infnorm_pos_le]
unfolding infnorm_def rsup_finite_ge_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image bex_simps
apply blast
by (rule abs_ge_zero)
from real_le_lsqrt[OF dot_pos_le th th1]
- show ?thesis unfolding real_vector_norm_def id_def .
+ show ?thesis unfolding real_vector_norm_def id_def .
qed
(* Equality in Cauchy-Schwarz and triangle inequalities. *)
@@ -5037,7 +5037,7 @@
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
hence "norm x \<noteq> 0" "norm y \<noteq> 0"
by simp_all
- hence n: "norm x > 0" "norm y > 0"
+ hence n: "norm x > 0" "norm y > 0"
using norm_ge_zero[of x] norm_ge_zero[of y]
by arith+
have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
@@ -5058,7 +5058,7 @@
lemma collinear_empty: "collinear {}" by (simp add: collinear_def)
-lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
+lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
apply (simp add: collinear_def)
apply (rule exI[where x=0])
by simp
@@ -5075,20 +5075,20 @@
lemma collinear_lemma: "collinear {(0::real^'n),x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *s x)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
- {assume "x=0 \<or> y = 0" hence ?thesis
+ {assume "x=0 \<or> y = 0" hence ?thesis
by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
{assume h: "?lhs"
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *s u" unfolding collinear_def by blast
from u[rule_format, of x 0] u[rule_format, of y 0]
- obtain cx and cy where
+ obtain cx and cy where
cx: "x = cx*s u" and cy: "y = cy*s u"
by auto
from cx x have cx0: "cx \<noteq> 0" by auto
from cy y have cy0: "cy \<noteq> 0" by auto
let ?d = "cy / cx"
- from cx cy cx0 have "y = ?d *s x"
+ from cx cy cx0 have "y = ?d *s x"
by (simp add: vector_smult_assoc)
hence ?rhs using x y by blast}
moreover